Proceedings ArticleDOI
Manifold Curvature From Covariance Analysis
Javier Álvarez-Vizoso,Michael Kirby,Chris Peterson +2 more
- pp 353-357
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TLDR
The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases.Abstract:
Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a high-dimensional cylinder orthogonal to the tangent space at the point cuts out a path-connected patch whose point-set distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the Frenet-Serret frame for curves, and to what we call the Ricci-Weingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases.read more
Citations
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Manifold curvature learning from hypersurface integral invariants
TL;DR: In this article, integral invariants obtained from principal component analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms.
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Geometry of curves in Rn from the local singular value decomposition
TL;DR: In this paper, the authors established a connection between the local singular value decomposition and the geometry of n-dimensional curves by linking the left singular vectors to the Frenet-Serret frame, and the generalized curvatures to the singular values.
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Local eigenvalue decomposition for embedded Riemannian manifolds
TL;DR: In this article, it was shown that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, has asymptotic expansions in terms of the mean and scalar curvatures.
Feature sensitive multiscale editing on surfaces
TL;DR: In this article, the surface features are encoded in a finite element matrix, passed to an algebraic multigrid (AMG) algorithm, which generates a matrix hierarchy ranging from fine to coarse representations of the initial fine grid matrix.
References
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A comprehensive introduction to differential geometry
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Proceedings ArticleDOI
Surface reconstruction from unorganized points
TL;DR: A general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points that is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
Journal ArticleDOI
The Rotation of Eigenvectors by a Perturbation. III
Chandler Davis,William Kahan +1 more
TL;DR: In this article, the difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other, and Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation or a computable residual.
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